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The k-deck of a graph is the multiset of its subgraphs induced by k vertices. A graph or graph property is l-reconstructible if it is determined by the deck of subgraphs obtained by deleting l vertices. We show that the degree list of an n-vertex graph is 3-reconstructible when n ≥ 7, and the threshold on n is sharp. Using this result, we show that when n ≥ 7 the (n − 3)-deck also determines whether an n-vertex graph is connected; this is also sharp. These results extend the results of Chernyak and Manvel, respectively, that the degree list and connectedness are 2-reconstructible when n ≥ 6, which are also sharp.MSC Codes: 05C60, 05C07

We study 2k-factors in (2r+1)-regular graphs. Hanson, Loten, and Toft proved that every (2r + 1)-regular graph with at most 2r cut-edges has a 2-factor. We generalize their result by proving for k ≤ (2r +1)/3 that every (2r +1)-regular graph with at most 2r − 3(k − 1) cut-edges has a 2k-factor. Both the restriction on k and the restriction on the number of cut-edges are sharp. We characterize the graphs that have exactly 2r − 3(k − 1) + 1 cut-edges but no 2k-factor. For k > (2r + 1)/3, there are graphs without cut-edges that have no 2k-factor, as studied by Bollobás, Saito, and Wormald.

A hypergraph $\mathcal H$ is super-pancyclic if for each $A \subseteq V(\mathcal H)$ with $|A| \geqslant 3$, $\mathcal H$ contains a Berge cycle with base vertex set $A$. We present two natural necessary conditions for a hypergraph to be super-pancyclic, and show that in several classes of hypergraphs these necessary conditions are also sufficient. In particular, they are sufficient for every hypergraph $\mathcal H$ with $ \delta(\mathcal H)\geqslant \max\{|V(\mathcal H)|, \frac{|E(\mathcal H)|+10}{4}\}$. We also consider super-cyclic bipartite graphs: those are $(X,Y)$-bigraphs $G$ such that for each $A \subseteq X$ with $|A| \geqslant 3$, $G$ has a cycle $C_A$ such that $V(C_A)\cap X=A$. Such graphs are incidence graphs of super-pancyclic hypergraphs, and our proofs use the language of such graphs.

A graph is ℓ-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting ℓ vertices. We prove that 3-regular graphs are 2-reconstructible.